Imaging using metamaterials

ABSTRACT

A system for imaging an object is provided. The system includes a light source configured to emit light having a predetermined wavelength towards the object. The system further includes a metalens including a metallic film having a plurality of slits defined therethrough, the plurality of slits having a width a and a periodicity d that are both less than the predetermined wavelength, wherein the object is positioned between the light source and the metalens. The system further includes a detector configured to acquire measurements indicative of light transmitted through the metalens, and a computing device communicatively coupled to the detector and configured to reconstruct an image of the object based on the acquired measurements.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of U.S. patent applicationSer. No. 13/716,531 filed Dec. 17, 2012, which claims the benefit ofU.S. Provisional Application No. 61/577,760 filed Dec. 20, 2011, both ofwhich are hereby incorporated by reference in their entirety.

BACKGROUND

Nonlinear materials give rise to a multitude of optical phenomena thathave important applications in technology and fundamental science. Oneexample is wave mixing, in which light of two different frequencies canbe added or subtracted through nonlinear optical responses to createlight at new frequencies. Wave mixing is an important optical processsince it is able to generate light at frequencies that are not availablein typical lasers and/or frequencies for which efficient photodetectorsare available. Another example is optical bistability, in which theintensity of output light can take two distinct stable values for agiven input, creating an optical two-state system. Bistable devices,such as optical logic gates and memory, are important for opticalcomputing, which has the potential for much faster computation speedsthan those in current computation devices. Yet another example isself-focusing, in which a light pulse through an optical fiber withsuitable nonlinearity can maintain its shape when propagating a longdistance in the fiber. Self-focusing is important for long-distancetelecommunications.

The degree of optical nonlinearity in a material depends upon thestrength of the optical field, and varies across different materials.For a nonlinear material of the Kerr-type, the relative permittivity, ∈,is dependent upon the electric field, E, as expressed by∈(E)=∈_(l)+χ⁽³⁾|E|², where ∈_(l) is the linear relative permittivity ofthe material and χ⁽³⁾ is the third-order nonlinear coefficient of thematerial. The nonlinear contribution to optical processes becomessignificant when χ⁽³⁾|E|² is of the order of ∈_(l), which is generallyrealized with a relatively strong electric field, as the third-ordernonlinear coefficient is relatively small in naturally occurring opticalmaterials. Accordingly, intense laser light is typically needed toobserver nonlinear optical phenomena, limiting the application ofnonlinear optics.

At least some nonlinear optical components utilize nonlinear resonatorsfor switching and modulation. However, a modulation speed and anavailable fractional bandwidth of known high quality-factor (high-Q)nonlinear resonators are reduced by the relatively large Q values.

For at least some known imaging applications, the diffraction limitrestricts the resolution of conventional microscopy to no less than halfan operating wavelength, as evanescent waves that carry subwavelengthinformation decay exponentially. By measuring an evanescent fielddirectly, near-field scanning optical microscopy (NSOM) exhibits a highresolution beyond the diffraction limit. The resolution of NSOM, whichdepends on the size of the aperture regardless of the operatingwavelength, has been demonstrated down to 20 nanometers (nm). However,for an aperture with size r in an infinitely thin film made of perfectmetal, the transmitted power is proportional to (r/λ)⁴, where λ is thewavelength of the normal incident light. For a relatively small aperturein a metal with finite thickness and conductance the transmitted poweris even weaker.

To enhance the transmitted power, various possible designs with a singleaperture have been proposed, ranging from periodic corrugations to theC-shape apertures, which have been shown to enhance the transmissionefficiency by two to three orders of magnitude. Further, subwavelengthperiodic apertures have been demonstrated to achieve improvedtransmission for potential applications for near field microscopy.However, at least some known strongly coupled periodic apertures existin such a configuration that each measurement contains the informationfrom all the apertures. Therefore, decomposing the coupled measurementsto reconstruct an image may be relatively difficult.

BRIEF DESCRIPTION

In one aspect, a system for imaging an object is provided. The systemincludes a light source configured to emit light having a predeterminedwavelength towards the object. The system further includes a metalensincluding a metallic film having a plurality of slits definedtherethrough, the plurality of slits having a width a and a periodicityd that are both less than the predetermined wavelength, wherein theobject is positioned between the light source and the metalens. Thesystem further includes a detector configured to acquire measurementsindicative of light transmitted through the metalens, and a computingdevice communicatively coupled to the detector and configured toreconstruct an image of the object based on the acquired measurements.

In another aspect, a method for imaging an object is provided. Themethod includes positioning the object between a light source and ametalens, wherein the metalens includes a metallic film having aplurality of slits defined therethrough, the plurality of slits having awidth a and a periodicity d. The method further includes emitting lightfrom the light source towards the object, wherein the emitted light hasa predetermined wavelength greater than the width a and the periodicityd. The method further includes acquiring, using a detector, measurementsindicative of light transmitted through the metalens, andreconstructing, using a computing device communicatively coupled to thedetector, an image of the object based on the acquired measurements.

In yet another aspect, a method for determining a refractive indexprofile of an object is provided. The method includes positioning theobject in front of a metalens that includes a plurality of slits definedtherethrough, emitting light having a predetermined wavelength towardsthe object and the metalens, acquiring, using at least one detector, aplurality of electric field measurements at a plurality of positions ofthe object relative to the metalens, and determining, using a computingdevice coupled to the at least one detector, the refractive indexprofile of the object based on the acquired electric field measurements.

BRIEF DESCRIPTION OF THE DRAWINGS

The embodiments described herein may be better understood by referringto the following description in conjunction with the accompanyingdrawings.

FIG. 1 is a schematic diagram of a two-dimensional array having anenhanced nonlinear response to light.

FIG. 2 is a unit cell representation of the array shown in FIG. 1.

FIG. 3 is a unit cell representation of an effective uniform dielectricslab.

FIG. 4 is a schematic diagram of a setup that may be used to simulatethe array shown in FIG. 1.

FIG. 5 is a graph illustrating transmission of incident light versus apermittivity ratio.

FIG. 6 is a graph illustrating transmission of incident light versus apermittivity ratio.

FIG. 7 is a graph illustrating transmission of incident light versus alength ratio.

FIG. 8 is a graph illustrating transmission of incident light versus alength ratio.

FIG. 9 is a graph illustrating total transmission of light versus apermittivity ratio.

FIG. 10 is a three-dimensional structure that has an enhanced nonlinearresponse to light.

FIG. 11 is a three-dimensional cube that has an enhanced nonlinearresponse to light.

FIG. 12 is a flowchart of an exemplary method of assembling ametamaterial having an enhanced nonlinear response to light.

FIG. 13 is a graph of a second harmonic transmission of a gratingdesigned for CuCl.

FIG. 14 shows a graph of a third harmonic transmission of a gratingdesigned for GaAs.

FIG. 15 shows a graph of a bistability curve of a metal-dielectricgrating containing Si.

FIG. 16 shows a graph of second harmonic generation of a grating withsilver and CuCl.

FIG. 17 is a schematic diagram of an imaging system.

FIG. 18 is a schematic diagram for explaining data acquisition and imagereconstruction in the system shown in FIG. 17.

FIG. 19 shows experimental results obtained using the system shown inFIG. 17.

FIG. 20 shows experimental results obtained using the system shown inFIG. 17.

FIG. 21 is a schematic diagram of a waveguide that may be used with thesystem shown in FIG. 17.

FIG. 22 shows experimental results obtained using the system shown inFIG. 17.

FIG. 23 shows experimental results obtained using the system shown inFIG. 17.

FIG. 24 is a block diagram of a computing device that may be used withthe system shown in FIG. 17.

DETAILED DESCRIPTION

Embodiments provide a system for imaging an object using a metalens. Themetalens includes a two-dimensional array including a metallic filmhaving periodic slits. The metalens exhibits artificial dielectricbehaviors and can be precisely mapped into a homogeneous dielectricslab. Further, the periodic structure results in a relatively hightransmission of incident light. Using the metalens, reconstructed imageshave a resolution much smaller than a wavelength of the incident light.

FIG. 1 is a schematic diagram of a two-dimensional array 100 having anenhanced nonlinear response to light. Array 100 includes a film 102 thatincludes a first surface 104 and an opposite second surface 106. Athickness, L, of film 102 is defined as the distance between firstsurface 104 and second surface 106.

As shown in FIG. 1, a plurality of slits 108 are defined through film102 extending from first surface 104 to second surface 106. Slits 108are each defined by a first edge 110 and an opposite second edge 112. Awidth, a, of each slit 108 is defined as the distance between first edge110 and second edge 112. Slits 108 in array 100 have a periodicity, d.Periodicity d is defined as the distance from a first edge 110 of a slit108 to a first edge 110 of an adjacent slit 108. As shown in FIG. 1,d>a.

Film 102 is composed of a metallic material. For example, film 102 maybe composed of gold, silver, aluminum, and/or any other metal thatenables array 100 to function as described herein. In the exemplaryembodiment, metallic material has a thickness greater than a skin depthof the material. Array 100 further includes a plurality of dielectricelements 114 that are located in slits 108. Specifically, for each slit108, dielectric element 114 is positioned between first edge 110 andsecond edge 112 and extends from first surface 104 to second surface106. Dielectric elements 114 are composed of a nonlinear material havinga relative permittivity, c. Dielectric elements 114 may be composed of,for example, fused silica having a nonlinear index of 3×10⁻¹⁶ cm²/W.

For Kerr-type nonlinear materials, such as dielectric elements 114 therelative permittivity can be expressed as a function of an electricfield, E, as ∈(E)=∈_(l)+χ⁽³⁾|E|², where ∈_(l) is the linear relativepermittivity of dielectric element 114 and χ⁽³⁾ is the third-ordernonlinear coefficient of dielectric element 114.

While in the exemplary embodiment, dielectric element 114 is a Kerr-typenonlinear material that exhibits third order nonlinearity, the systemsand methods described herein may also be implemented using Pockels-typenonlinear materials that exhibit second order nonlinearity. For aPockels-type dielectric element, the relative permittivity can beexpressed as ∈(E)=∈_(l)+χ⁽²⁾|E|, where ∈_(l) is the linear relativepermittivity of the dielectric element and χ⁽²⁾ is the second-ordernonlinear coefficient of the dielectric element.

Generally, the order of magnitude of χ⁽³⁾ in units of centimeterssquared per Farad squared (cm²/F²) is 10⁻¹⁶ to 10⁻¹⁴ for glasses, 10⁻¹⁴to 10⁻¹⁷ for doped glasses, and 10⁻¹⁰ to 10⁻⁸ for organic materials.Accordingly, as χ⁽³⁾ is generally relatively small, the nonlinearcomponent of c becomes significant when the applied electric field E isrelatively large. Accordingly, without a substantially large electricfield, the practical application of nonlinear properties of naturallyoccurring nonlinear materials is fairly limited. However, using themetamaterials described herein, a nonlinear response to light can begenerated at a lower electric field, as described herein.

Given the configuration of array 100, there exists a transverseelectromagnetic (TEM) mode between slits 108 for incident light having atransverse magnetic (TM) polarization. The TEM mode allows fortransmission of light through slits 108 when the periodicity, d, is muchsmaller than the wavelength of the incident light, λ. Under theseconditions, the light squeezes through slits 108, resulting in asignificant increase in the field intensity, E, within slits 108. Theamplified field intensity causes dielectric elements 114 to experience alarger intensity than the incident field, such light transmitted throughsilts 108 has a larger nonlinear response. Accordingly, array 100exhibits an enhanced nonlinear response.

To understand the enhanced nonlinearity of array 100, array 100 can bemathematically mapped to an effective uniform dielectric slab. That is,array 100 responds to light in the same way that an effective uniformdielectric slab would respond. Array 100 is mapped to the effectiveuniform dielectric slab using a unit cell correspondence, in whichpotential difference, instantaneous power flow, and energy between twoequivalent unit cells are conserved.

FIG. 2 is a unit cell 200 of array 100. FIG. 3 is a unit cell 300 of auniform dielectric slab. Unit cell 200 and unit cell 300 each have awidth of periodicity d. Unit cell 200 has a thickness L, and unit cell300 has a thickness L. For the purposes of mapping unit cell 200 to unitcell 300, it is assumed that film 102 allows no field penetration (i.e.,light only passes through slit 108).

The potential difference over unit cell 200 is a·E, as the field cannotpenetrate film 102. The potential difference over unit cell 300 is d·Ē,where E is the electric field in unit cell 200, and Ē is the electricfield in unit cell 300. Setting the potential differences of unit cell200 and unit cell 300 equal to each another, the electric field scalingcondition becomes

$\overset{\_}{E} = {{E\left( \frac{a}{d} \right)}.}$

The instantaneous power flow over unit cell 200 is (E×H)×a, and theinstantaneous power flow over unit cell 300 is (Ē× H)×d, where H is themagnetic field in unit cell 200, and H is the magnetic field in unitcell 300. Setting the instantaneous power flows of unit cell 200 andunit cell 300 equal to each other, and using

${\overset{\_}{E} = {E\left( \frac{a}{d} \right)}},$the magnetic field scaling condition becomes H=H.

The energy over unit cell 200 is

${\frac{1}{2}\left( {{ɛ\; E^{2}} + {\mu\; H^{2}}} \right) \times L \times a},$where ∈ is the relative permittivity of dielectric element 114, and μ isthe permeability of dielectric element 114. The energy over unit cell300 is

${\frac{1}{2}\left( {{\overset{\_}{ɛ\; E}}^{2} + {\overset{\_}{\mu\; H}}^{2}} \right) \times \overset{\_}{L} \times d},$where ∈ is the relative permittivity of the dielectric slab and μ is thepermeability of the dielectric slab. Assuming that μ=μ=1 and

${\overset{\_}{L} = {L\left( \frac{a}{d} \right)}},$and using the scaling conditions above, when the energies of unit cell200 and unit cell 300 are set equal to each other, the permittivityscaling condition becomes

$\overset{\_}{ɛ} = {{ɛ\left( \frac{d}{a} \right)}^{2}.}$Accordingly,

${\overset{\_}{ɛ} = {{\left( \frac{d}{a} \right)^{2}\left( {ɛ_{l} + {\chi^{(3)}{E}^{2}}} \right)} = {{\overset{\_}{ɛ}}_{l} + {{\overset{\_}{\chi}}^{(3)}{\overset{\_}{E}}^{2}}}}},$where ∈_(l) is the linear relative permittivity of unit cell 300, and χ⁽³⁾ is the third-order nonlinear coefficient of unit cell 300.

From these equations, it can be determined that

${\overset{\_}{ɛ}}_{l} = {ɛ\left( \frac{d}{a} \right)}^{2}$and

${\overset{\_}{\chi}}^{(3)} = {{\chi^{(3)}\left( \frac{d}{a} \right)}^{4}.}$Performing similar calculations for a Pockels-type material, it can bedetermined that

${\overset{\_}{\chi}}^{(2)} = {{\chi^{(2)}\left( \frac{d}{a} \right)}^{3}.}$Thus, the effective third-order nonlinear coefficient χ ⁽³⁾ of array 100is the third-order nonlinear coefficient of dielectric element 114amplified by a factor of

$\left( \frac{d}{a} \right)^{4}.$As such, by manipulating the ratio of d to a in array 100, the nonlinearproperties of dielectric element 114 can be significantly enhanced. Forexample, when the ratio of d to a is 4, the third-order nonlinearcoefficient of array 100 is 256 times the third-order nonlinearcoefficient of dielectric element 114.

The permittivity scaling condition

$\overset{\_}{ɛ} = {ɛ\left( \frac{d}{a} \right)}^{2}$is asymptotically exact at deep subwavelength scales, λ/d>>1. In someembodiments, to extend this to all subwavelength scales, a generalscaled mapping is derived using small refinements to the simple scaledmapping as

${{\overset{\_}{ɛ}}_{l} = {{ɛ_{c}\left( \frac{d}{a} \right)}^{2}ɛ_{l}}},$

${\overset{\_}{\chi}}^{(2)} = {{\chi_{c}^{(2)}\left( \frac{d}{a} \right)}^{3}\chi^{(2)}}$for second order, and

${\overset{\_}{\chi}}^{(3)} = {{\chi_{c}^{(3)}\left( \frac{d}{a} \right)}^{4}\chi^{(3)}}$for third order, where the dielectric refinements are ∈_(c), χ_(c) ⁽²⁾,and χ⁽³⁾. The values of the dielectric refinements are calculated fromthe linear solutions of the dielectric grating. The linear dielectricrefinement ∈_(c) is calculated by matching the linear permittivity ofthe simple scaled mapping to yield identical transmission coefficientsof the metal-dielectric grating, and solved using

$T_{grating}^{- 1} = {1 + {\frac{\left( {{\left( {d/a} \right)^{2}ɛ_{l}ɛ_{c}} - 1} \right)^{2}}{4\left( {d/a} \right)^{2}ɛ_{l}ɛ_{c}}{{\sin^{2}\left( {\frac{2\pi}{\lambda}{\overset{\_}{L}\left( {d/a} \right)}\sqrt{ɛ_{l}ɛ_{c}}} \right)}.}}}$The nonlinear dielectric refinements are calculated by dividing theelectric fields within the slit by the fields with corrected linearpermittivity as

${\chi_{c}^{(2)} = \left( \frac{E}{\left( {d/a} \right)E_{c}} \right)^{2}}\;$for second order and

${\chi_{c}^{(3)} = \left( \frac{E}{\left( {d/a} \right)E_{c}} \right)^{4}}\;$for third-order nonlinearity. Calculating the refinements yields anexact mapped optical response of the grating. These refinements aregenerally very small, on the order of 1 percent, and produce the sameorder of magnitude enhancement for nonlinear materials. When ∈_(c) issmall, χ_(c) ⁽²⁾ and χ_(c) ⁽³⁾ are approximately 1. By calculating therefinements, a general scaled mapping is produced with full accuracyover all subwavelength scales.

Notably, for a given value of d/a, array 100 will not exhibit anenhanced nonlinear response in the transmission direction (i.e. througharray 100) for all frequencies of incident light. Specifically, atcertain frequencies, the transmission of light through array 100 isrelatively small, and most of the incident light is reflected by array100. At such frequencies, array 100 will not exhibit an enhancednonlinear response. Accordingly, the non-linear response of array 100 isdependent at least in part on the frequency of the incident light. Inthe exemplary embodiment, the frequency of the incident light is 1.55microns. Alternatively, incident light may have any frequency thatenables array 100 to function as described herein.

To demonstrate the enhanced nonlinear response of array 100, lighttransmission through array 100 and the effective dielectric slab wassimulated. The first simulation demonstrated the mapping in wave mixingtransmission, specifically third harmonic generation, in which a singlefrequency generates new light at three times the incident frequencythrough frequency summing within the nonlinear material. This wassimulated over many incident intensities. The second simulationdemonstrated variation in the transmission of third harmonic generationover a range of film thicknesses, while keeping a constant incidentintensity. The third simulation demonstrated the mapping in opticalbistability. From the results of each simulation, the robustness of themapping between array 100 and the effective dielectric slab wasdemonstrated for different film thicknesses, intensities, andwavelengths.

FIG. 4 is a schematic diagram of a setup 400 that was used to performthe simulations for array 100. The periodic system geometry and materialdefinitions were defined at the center of the computational domain, andfilm 102 ran along the center of the horizontal axis. The width of thecomputational domain was one full periodicity, d, of the periodic systemgeometry, with slit 108 and dielectric element 114 centered within film102. Since the computational domain was symmetric about the verticalaxis, a symmetry condition was applied along that axis to reducecomputational work by half.

To eliminate reflections at the top and bottom of the computationaldomain, PML (perfectly-match-layers) 402, near-perfect absorbers ofelectromagnetic waves, were applied at the top and bottom boundary endsof the computational domain. A thickness of PMLs 402 was adjusted toone-half of the incident source wavelength, λ. A periodic boundarycondition 404 was applied to a left edge and a right edge of thecomputational domain, although all simulations were performed with lightat normal incidence to the film 102.

A plane-wave source 406 was used to generate light at a singlewavelength, λ, and intensity, and was located just below the top PML402. The distance between source 406 and the film 102 was twowavelengths. The electric and magnetic field data was collected at adata collection point 408 just above the bottom PML 402. The distancebetween data collection point 408 and film 102 was also two wavelengths.In total, the length of the computational domain was five wavelengthsplus the film thickness, L. A similar setup was used for the uniformdielectric slab, except that only one spatial dimension was used, due tothe uniformity of the slab across the horizontal axis. This reduced thecomputational time of the effective slab results.

The first simulation of the mapping between array 100 and the effectivedielectric slab demonstrated the transmission of incident and thirdharmonic light over a range of intensities at normal incidence to film102. Since the index of refraction, n, of nonlinear optical materials isintensity dependent, the transmission of both the first and thirdharmonic frequency was expected to vary with intensity. In thissimulation, array 100 had the following parameters: λ=12, d/a=4,n_(o)=1.5, a=0.1, λ/d=30, L=35.2, and χ⁽³⁾=3.9×10⁻⁵, where n_(o) is theindex of refraction of dielectric element 114 (note these are in scaleinvariant units). Using the mapping from the above equations, theeffective slab had the following parameters: λ=12, ∈ _(l)=36, L=8.8, andχ ⁽³⁾=0.01.

FIG. 5 is a graph 500 illustrating transmission of incident light versusa permittivity ratio. FIG. 6 is a graph 600 illustrating transmission ofthird harmonic light versus a permittivity ratio. The transmission ofthe incident and third harmonic light was quantified using transmissioncoefficients defined as follows,

$T_{\omega} = \frac{{E_{\omega}}^{2}}{{E_{0}}^{2}}$for the incident frequency, and

$T_{3\omega} = \frac{{E_{3\omega}}^{2}}{{E_{0}}^{2}}$for the third harmonic frequency. For graphs 500 and 600, the pointsrepresent the calculated transmission coefficients for array 100, andthe line represents the calculated transmission coefficients or theeffective dielectric slab. The transmission coefficients were plottedversus the permittivity ratio,

$\frac{\chi^{(3)}{E_{0}}^{2}}{ɛ_{l}},$which expresses the magnitude of the nonlinear part of the totalpermittivity χ⁽³⁾|E₀|², over the linear part, ∈_(l). Accordingly, thepermittivity ratio varies linearly with incident intensity |E₀|².

Graphs 500 and 600 demonstrate significant variation in transmissionwith incident intensity. The transmission at the incident and thirdharmonic frequencies increases as the intensity is increased until 0.14permittivity ratio is reached, where both frequencies have a maximumtransmission of 0.75 and 0.06 respectively. Past 0.14, the transmissioncoefficients decrease. At 0.40, however, the transmission of thirdharmonic increases once again, whereas the transmission of incidentlight continues to decrease.

For both the incident and third harmonic light, the transmission ofarray 100 substantially agrees with the mapping to the transmission ofthe effective dielectric slab. Slight differences in the finalcalculations are a result of the fundamental assumption that array 100experiences deep subwavelength resonance. For larger λ/d, this errorwould be reduced. Thus, the first simulation establishes that array 100acts as a uniform dielectric slab with an index of refraction of 6 andnonlinear index of 0.01, which results in an overall increase of 4 inthe index of refraction and an increase of 256 in the third-ordernonlinear coefficient of dielectric element 114.

The second simulation of the mapping between array 100 and the effectivedielectric slab demonstrated the transmission of the incident and thirdharmonic light over a range of film thicknesses L, while keeping thefield intensity fixed at a normal incidence to film 102. Thetransmission at both the incident and third harmonic frequencies wasexpected to vary with film thickness, as wave mixing is very sensitiveto variations in film thickness variation. In this simulation, array 100had the following parameters: λ=16, d/a=8, n_(o)=1.5, a=0.1, |E₀|=200,λ/d=20, L₀=32/3, and χ⁽³⁾=2.44×10⁻⁶ (note these are in scale invariantunits). Using the mapping from the above equations, the effective slabhad the following parameters: λ=16, ∈ _(l)=144, L ₀=4/3, and χ ⁽³⁾=0.01.

FIG. 7 is a graph 700 illustrating transmission of incident light versusa length ratio. FIG. 8 is a graph 800 illustrating transmission of thirdharmonic light versus a length ratio. The transmission of the incidentand third harmonic light were quantified using transmission coefficientsdefined as in the first simulation. For graphs 700 and 800, the pointsrepresent the calculated transmission coefficients for array 100, andthe line represents the calculated transmission coefficients of theeffective dielectric slab. The transmission coefficients were plottedversus length ratio, L/L₀, which expresses the film thickness L over thedefined film thickness L₀.

Graphs 700 and 800 both display a multitude of nonlinear optical effectswith film thickness. The transmission of the incident and third harmonicfrequencies are periodic, with a transmission periodicity of 0.45 inlength ratio. The transmission at the incident frequency exhibitsbistability through discrete jumps at length ratios of 1.3 and 1.8.Further simulation near these length ratios would likely yield a secondstable transmission branch. The transmission at the third harmonicfrequency shows a much more complicated response to film thickness thanat the incident frequency. At length ratios of 1.1 and 1.6, transmissionsharply peaks, whereas the corresponding transmission at the incidentfrequency is near minimum. Two smaller regions of increased thirdharmonic generation occur at length ratios from 1.3-1.4 and 1.75-1.85.

As in the first simulation, for both the incident and third harmoniclight, the transmission of the array 100 substantially agrees with themapping to the transmission of the effective dielectric slab. Slightdifferences in the final calculations are a result of the fundamentalassumption that array 100 experiences deep subwavelength resonance. Forlarger λ/d, this error would be reduced. Thus, the second simulationestablishes that array 100 acts as a uniform dielectric slab with anindex of refraction of 12 and nonlinear index of 0.01, which results inan overall increase of 8 in the index of refraction and an increase of4096 in the third-order nonlinear coefficient of dielectric element 114.

The third simulation of the mapping between array 100 and the effectivedielectric slab demonstrated transmission exhibiting bistability over arange of intensities at normal incidence to film 102. Both array 100 andthe effective dielectric slab were expected to display two distinctregions of stable transmission. In this simulation, array 100 had thefollowing parameters: λ=8, d/a=4, n_(o)=1.5, a=0.1, λ/d=20, L=64/3, andχ⁽³⁾=3.9×10⁻⁵ (note these are in scale invariant units). Using themapping from the above equations, the effective slab had the followingparameters: λ=8, ∈ _(l)=36, L=16/3, and χ ⁽³⁾=0.01.

FIG. 9 is a graph 900 illustrating total transmission of light versusthe permittivity ratio. The total transmission was quantified using thetransmission coefficient, defined as follows in Equation (1):

$\begin{matrix}{T_{TOTAL} = {\frac{\sum{E_{i \times \omega}}^{2}}{{E_{0}}^{2}}.}} & (1)\end{matrix}$

To calculate the upper branch of the bistability curve, the intensity ofthe plane-wave source increased from zero to a steady-state value. Thisis essentially the same as the previous two simulations, except thatthere was a slower rise in the steady-state intensity. To calculate theupper branch, a plane-wave source at a steady-state intensity was raisedto twice its value using a temporal Gaussian pulse with the peakintensity occurring at the frequency of interest. Using this source, thetemporal Gaussian pulse pushed film 102 into a higher-intensity regionof single-valued transmission beyond the bistable region. Through thepulse decay, the film fell back into the region of bistability at asteady-state intensity of interest. However, having “remembered” comingfrom a region of higher intensity, through nonlinear optical processes,the transmission jumped down to the lower branch. Using both sources,the lower and upper branches of bistability were successfullydemonstrated.

In graph 900, the points represent the calculated total transmission ofthe upper branch of array 100 using the single plane-wave source, whilethe crosses represent the calculated total transmission of the lowerbranch of array 100 using the dual plane-wave, Gaussian pulsed source.The solid line represents the calculated total transmission of the upperbranch of the effective dielectric slab, and the dashed line representsthe calculated total transmission of the lower branch of the effectivedielectric slab.

In graph 900, the total transmission is single-valued for permittivityratios from 0 to 1.8. This is confirmed for both the single plane-waveand the dual plane-wave, Gaussian pulsed source, as both resulted in thesame total transmission response. At 1.85, the total transmission jumpsfrom 0.10 to 0.40 when the intensity is raised from zero, hence markingthe beginning of the upper branch. However, when the intensity isincreased past 2.10 and lowered back to 1.85, the total transmissiondoes not jump, hence marking the beginning of the lower branch. At 2.10,the total transmission of the lower branch jumps from 0.15 to 0.35. Past2.10, the total transmission becomes single-valued again, as confirmedfor both the single plane-wave and the dual plane-wave, Gaussian pulsedsource, as both resulted in the same total transmission response. Giventhese properties, the bistability region is referred to as clockwise, asthe intensity profile follows a clockwise path to produce a fullbistability curve.

The total transmission of array 100 agrees with the mapping to the totaltransmission of the effective dielectric slab. Slight differences in thefinal calculations are a result of the fundamental assumption that array100 experiences deep subwavelength resonance. For larger λ/d, this errorwould be reduced. Thus, the third simulation establishes that array 100acts as a uniform dielectric slab with an index of refraction of 6 andnonlinear index of 0.01, which results in an overall increase of 4 inthe index of refraction and an increase of 256 in the third-ordernonlinear coefficient of dielectric element 114.

FIG. 10 is a three-dimensional structure 1000 that has an enhancednonlinear response to light. Structure 1000 includes a first plate 1002and an opposite second plate 1004 that are coupled to one another by ashaft 1006. First plate 1002 and second plate 1004 are substantiallyparallel to one another, and shaft 1006 is substantially orthogonal toplates 1002 and 1004. Plates 1002 and 1004 each have a first surface1007, an opposite second surface 1008, and a thickness L. Plates 1002and 1004 each include a plurality of substantially parallel arms 1010extending from a crossbeam 1012. A plurality of slits 1014 are formedbetween arms 1010 and extending from first surface 1007 to secondsurface 1008.

In the embodiment shown in FIG. 10, plates 1002 and 1004 each includesix arms 1010 and four slits 1014. However, plates 1002 and 1004 mayinclude any number of arms 1010 and/or slits 1014 that enable structure1000 to function as described herein. Similar to slits 108, slits 1014have a width, a, and a periodicity, d. In structure 1000, each slit 1014includes a dielectric element 1016 positioned in slit 1014.

Similar to film 102, plates 1002 and 1004 are composed of a metallicmaterial. For example, plates 1002 and 1004 may be composed of silver,gold, aluminum, and/or any other suitable material having a thicknessgreater than a skin depth of the material. Dielectric elements 1016 maybe composed of fused silica having a nonlinear index of 3×10⁻¹⁶ cm²/W.Plates 1002 and 1004 and dielectric elements 1016 exhibit an enhancednonlinear response to light, similar to film 102 and dielectric elements114. Accordingly, similar to array 100, the ratio of d to a can bemanipulated in structure 1000 to alter the nonlinear response ofstructure 1000, similar to array 100.

FIG. 11 is a three-dimensional cube 1100 having an enhanced nonlinearresponse to light. Cube 1100 may be formed by coupling additional plates1102 to structure 1000, such that additional plates 1102 aresubstantially orthogonal to plates 1002 and 1004. Accordingly, firstplate 1002, second plate 1004, and additional plates 1102 form the sixfaces of cube 1100.

Additional plates 1102 are substantially similar to plates 1002 and 1004of structure 1000. That is, additional plates 1102 include dielectricelements 1016 positioned within slits 1014 in plates 1102. In theembodiment shown in FIG. 11, each additional plate 1102 is coupled toshaft 1006 by a member 1104. Alternatively, additional plates 1102 arecoupled to structure 1000 by any means that enable cube 1100 to functionas described herein. As the six faces of cube 1100 each include slits1014 and dielectric elements 1016, to generate an enhanced nonlinearresponse, light can be transmitted towards cube 1100 from a variety ofdirections and/or angles.

FIG. 12 is a flowchart of an exemplary method 1200 of assembling ametamaterial having an enhanced nonlinear response to light, such asarray 100, structure 1000, and/or cube 1100. Method 1200 includesproviding 1202 a metallic film having a plurality of slits definedtherethrough, such as metallic film 102 and slits 108. The slits in themetallic film have a width a and a periodicity d. A plurality ofdielectric elements, such as dielectric elements 114 are inserted 1204into the slits. A relationship exists between a and d such that themetamaterial formed from the metallic film and the dielectric elementsexhibits an enhanced nonlinear response to light that is transmittedthrough the slits in the metamaterial.

Validations

Rigorous numerical calculations of the far-field transmission for boththe metal-dielectric grating and the general scaled mapping wereperformed. The electrodynamic calculations were processed using afinite-difference time-domain (FDTD) numerical method. The simulationused in all cases was MEEP. These examples demonstrate the enhancementof the grating and the robustness of the general scaled mapping throughfundamental nonlinear optical phenomena, specifically harmonicgeneration in second-order and third-order nonlinear materials, andoptical bistability. Reference slabs containing the nonlinear materialwithin the grating were plotted to show the enhanced response of thegrating. From the results of each case, the accuracy of the scaledmapping and the many orders of magnitude of enhancement of the gratingfor different grating thicknesses, periodicities, and intensities wereestablished. Although the grating produces an enhancement for anysuitable wavelength, the examples herein utilize the mid-infrared regionof light at approximately 10.6 μm.

The first example is the enhancement of harmonic generation. Forsecond-order non-linear materials, a harmonic at twice the inputfrequency is formed, and for third-order material, a harmonic at threetimes the input frequency is formed. Other higher harmonics are formed;however, these are much smaller compared to the second or thirdharmonics. To demonstrate second harmonic generation, we chose cuprouschloride (CuCl) as the second-order dielectric in the grating. At 10.6μm, CuCl is assumed to have a linear permittivity ∈_(l)=4 and asecond-order nonlinearity χ⁽²⁾=6.7 pm/V. For the grating to besubwavelength, the grating periodicity d=1.178 μm and the slit widtha=0.147 μm. The film thickness L=10.6 μm for the grating to be nearresonance. From the general scaled mapping, the grating acts as ahomogeneous dielectric slab with enhanced linear permittivity of ∈_(l)=258 and enhanced second order non-linearity χ ⁽²⁾=3444 pm/V at aslab thickness L=1.325 μm. In particular, CuCl experiences anenhancement of 64.5 in linear relative permittivity and 514 insecond-order susceptibility atop of its naturally occurringconstituents.

FIG. 13 shows a graph 1300 of the second harmonic transmission of thegrating designed for CuCl and the homogeneous dielectric slab from thegeneral scaled mapping for several input field strengths, E₀. The inputfield strength is presented as a ratio defined as

$\frac{\chi^{(2)}{E_{0}}}{ɛ_{l}}$for second order materials. This ratio scales the magnitude of thenonlinear part of the permittivity over the linear permittivity. Whenthe ratio is less than one, the linear part of the permittivitydominates, whereas when the ratio is greater than one, the nonlinearpart dominates. For all input fields, general scaled mapping fitsextremely well with the metal-dielectric grating. This result confirmsthe scaled mapping analysis for gratings with second-order materials.Also, this result establishes the general scaled mapping as a predictivemodel for second harmonic generation in metal-dielectric gratings.

In addition, the nonlinear optical response of several uniform slabscontaining CuCl at the same field inputs was calculated to verify theenhancement of the grating. To best illustrate the enhancement from thegrating, three different film thicknesses were compared: when referencethickness is equal to the grating thickness, the mapped thickness, andon resonance. For the reference slab, the thickness of the grating andresonance condition are equivalent. For the field strengths used in FIG.13, the ratio is relatively small. Under the low field input conditions,the second harmonic generation in both systems are in their beginningstages, where conversion efficiency is small. This is a good region tocompare the optical response of the grating against the reference slabsince this avoids areas of harmonic saturation. Because the conversionto second harmonic generation is linear at low field strength, greaterthan two orders of magnitude enhancement is clearly evident, and upholdsthe general scaled mapping prediction that the grating designedgenerates an enhancement of 514.

In demonstration of third harmonic generation, we chose gallium arsenide(GaAs) as the third-order dielectric in the grating. At 10.6 μm, GaAs isassumed to have a linear permittivity ∈_(l)=10.13 and a third-ordernonlinearity χ⁽³⁾=0.120 cm³/erg. For the grating to be subwavelength,the grating periodicity d=0.8 μm and the slit width a=0.1 μm. The filmthickness L=3.352 μm for the grating to be near resonance. From thegeneral scaled mapping, the grating acts as a homogeneous dielectricslab with enhanced linear permittivity of ∈ _(l)=644 and enhanced thirdorder non-linearity χ ⁽³⁾=491.5*10⁻¹⁰ cm³/erg at a slab thicknessL=0.419 μm. In particular, GaAs experiences an enhancement of 64.4 inlinear relative permittivity and 4096 in third-order susceptibility atopof its naturally occurring constituents.

FIG. 14 shows a graph 1400 of the third harmonic transmission of thegrating designed for GaAs and the homogeneous dielectric slab from thegeneral scaled mapping for several input field strength. The input fieldstrength is presented as a ratio defined as

$\frac{\chi^{(3)}{E_{0}}^{2}}{ɛ_{l}}$for third-order materials. For all input fields, the general scaledmapping fits the metal-dielectric grating. This result confirms thescaled mapping analysis for gratings with third-order materials. Also,this result establishes the general scaled mapping as a predictive modelfor third harmonic generation in metal-dielectric gratings.

In addition the nonlinear optical response of several uniform slabscontaining GaAs at the same field inputs was calculated to verify theenhancement of the grating. The same reference slab technique used inthe previous validation was used. Also, for the reference slab, thethickness of the grating and resonance condition are equivalent. For thefield strengths used in FIG. 14, the ratio is very small. For thesmallest ratios shown, the third harmonic generation in both systems isin their beginning stages, where conversion efficiency is small. Becausethe conversion to third harmonic generation is linear at these low fieldstrengths, three orders of magnitude enhancement is clearly evident, andupholds the general scaled mapping prediction that the grating designedgenerates an enhancement of 4096. For the larger ratios shown, thegrating appears to plateau. This is a result of large modification ofthe total permittivity such that the transmission of the fundamentalfrequency is pushed off resonance. If the third harmonic efficiency,defined as I_(3ω)/I_(ω), was shown, harmonic conversion would increaselinearly, and thus, demonstrate the orders of magnitude enhancementagainst the reference slabs at these larger ratios.

The second validation is the enhancement of far-field opticalbistability. For certain input field intensities, a nonlinear materialcan generate two distinct, stable output intensities, this is known asthe bistability region. The output with the greater transmittance is theupper curve, whereas the lower transmittance is the lower curve. Forcounterclockwise bistability, the lower curve is reached when drivenfrom intensities lower than the bistability region. Conversely, theupper curve is reached when driven from intensities higher than thebistability region. Because of this, the calculation of both branches ofbistability requires added computation finesse. Discrete jumps intransmittance occur when passing through the bistability region fromeither high or low single-state intensities. These discrete jumpscharacterize the bistability region. In addition, optical bistabilityonly occurs for specific film thicknesses in third-order nonlinearmaterials.

To demonstrate bistability, silicon (Si) was chosen as the third-orderdielectric in the grating. At 10.6 μm, Si is assumed to have a linearpermittivity ∈_(l)=12.094 and a third-order nonlinearityχ⁽³⁾=0.060*10⁻¹⁰ cm³/erg. For the grating to be subwavelength, thegrating periodicity d=0.884 μm and the slit width a=0.221 μm such thatd/a=4. Because bistability occurs at a specific film thickness, the filmthickness L=4.24 μm for the grating. At this film thickness, bistabilityoccurs for the metal-dielectric grating. From the general scaledmapping, the grating acts as a homogeneous dielectric slab with enhancedlinear permittivity of ∈ _(l)=194.75 and enhanced third ordernon-linearity χ ⁽³⁾=15.36*10⁻¹⁰ cm³/erg at a slab thickness L=1.06 μm.In particular, Si experiences an enhancement of 16.1 in linear relativepermittivity and 256 in third-order susceptibility atop of its naturallyoccurring constituents.

FIG. 15 shows a graph 1500 of the bistability curve of themetal-dielectric grating containing Si and the homogeneous dielectricslab from the general scaled mapping for several input field strengths.For all input fields, the general scaled mapping fits with themetal-dielectric grating. The scaled mapping not only predicts thecorrect bistable branch transmissions, but also the specific input fieldintensities at which the bistability region occurs, which is importantin the measurement and design of bistability in devices utilizing thephenomena. This result confirms the scaled mapping analysis for gratingswith third-order materials in bistability. Also, this result establishesthe general scaled mapping as a predictive model for bistability inmetal-dielectric gratings.

In addition, the bistability of a uniform slab containing Si at the samefield inputs was calculated to verify the enhancement of the grating. Tobe consistent, the reference slab thickness was chosen to be near thefirst onset of bistability to match the conditions of the grating. InFIG. 15, the metal-dielectric grating shows bistability at much lowerintensities relative to the reference slab. In particular, thebistability region of the grating occurs two orders of magnitude soonerthan the reference slab. This result upholds the general scaled mappingprediction that the grating designed generates an enhancement of 256.For fixed d/a, the intensities in bistability region of the grating canbe lowered with small increases in film thickness, to further lower itsonset. Also, increasing d/a would further lower the onset ofbistability. This result is crucial for low-powered devices utilizingoptical computation.

The last validation is the enhancement of far-field second harmonicgeneration gratings containing a lossy metal. To demonstrate this,silver was chosen for the metal gratings. For consistency with theprevious validations, the same second-order nonlinear material, cuprouschloride (CuCl) at 10.6 μm, was used. Designing the grating to besubwavelength, the grating periodicity d=5.3 μm and the slit widtha=1.325 μm, such that d/a=4. The film thickness L=2.49 μm for thegrating to be near resonance. Since the scaled mapping does not involvelossy metals, no scaled mapping is demonstrated herein. However, usingit as a guide for the level of enhancement, it is expected thattwo-orders of magnitude of enhancement should be observed for the chosenparameters.

FIG. 16 shows a graph 1600 of the second harmonic generation of thegrating with silver and CuCl. At the same field inputs, reference slabscontaining CuCl at the three previously proposed thicknesses were alsoanalyzed. Under the low field input conditions, the second harmonicgeneration in both systems is in their beginning stages, whereconversion efficiency is small. Because the conversion to secondharmonic generation is linear at low field strength, two orders ofmagnitude enhancement is evident, even with losses from silver gratings.For materials at longer wavelengths, losses from metal will diminish,where as shorter wavelengths will reduced the effectiveness of metal dueto larger field penetration into the metal. Overall, this resultdemonstrates the enhancement of nonlinear materials and justifiesprevious assumptions.

The metamaterials described herein (e.g., array 100 shown in FIG. 1,structure 1000 shown in FIG. 10, and cube 1100 shown in FIG. 11), inaddition to having an enhanced nonlinearity, are also capable ofoperating in a low-Q regime. In the low-Q regime, a cavity effect isminimized to yield relatively short intrinsic temporal response times,which may be as short as a few picoseconds.

For example, in one numerical simulation demonstrating second harmonicgeneration, for an array 100 with a=0.165 μm, d=1.325 μm (d/a=8), andL=11.2 μm, the Q-factor was 180. In another numerical simulationdemonstrating third harmonic generation, for an array 100 with a=0.294μm, d=1.767 μm (d/a=6), and L=2.648 μm, the Q-factor was 38. Further, aquality-factor value for the metal-dielectric grating containing Si(discussed above in relation to FIG. 15) at resonance was 380 (i.e.,Q=380).

From numerical simulation, it was also observed that a bistability ofthe metamaterials described herein may be switched between high and lowstates within ˜30 optical cycles, which corresponds to a switching timeof approximately 1 picosecond. Accordingly, by operating at a low-Qregime, a temporal response of the metamaterials described herein may beas short as a few picoseconds.

The metamaterials described herein may be implemented in imagingsystems. By using a subwavelength metal dielectric grating, opticalresolution beyond a diffraction limit may be achieved. Specifically, ametal dielectric grating, such as array 100 (shown in FIG. 1), functionsas a metalens. Metalens may also be referred to as an array ofcorrelated nano-torches (CNTs).

FIG. 17 is a schematic diagram of an imaging system 1700 that includesarray 100, which is also referred to herein as a metalens 100. Metalens100 has a resolution that is independent of an operating wavelength.Instead, the resolution is determined by the dielectric slit width, a.Accordingly, the resolution may be orders of magnitude smaller than theincident light wavelength, λ. In a transmission mode, system 1700operates with a relatively large transmitted signal, enabling far-fieldmeasurements. Further, by scaling dimensions of system 1700, awavelength to resolution ratio can be maintained through the optical tonear-infrared regime.

System 1700 includes a light source 1702 that emits light havingwavelength λ, in a k direction towards metalens 100. In the exemplaryembodiment, wavelength λ is significantly larger than slit width a. Anobject 1704 to be imaged is positioned between light source 1702 andmetalens 100. One or more detectors 1706 are positioned on an oppositeside of metalens 100 from object 1704. Detector 1706 detects lighttransmitted through metalens 100 to facilitate imaging object 1704.

In system 1700, the field strength in the dielectric slits 108 isenhanced, instead of suppressed, by the narrow slit width a. Toreconstruct an image of object 1704, an algorithm is used, as describedherein. Using system 1700, in one embodiment, for a wavelength of λ=2.08micrometers (μm), a resolution of 40 nm (˜λ/50) may be achieved. Such arelatively large wavelength to resolution ratio may be maintained in theentire optical spectrum range, and can be adjusted relatively easily bychanging dimensions of metalens 100 with respect to changes inwavelength λ. By exploiting the relatively large transmitted signal frommetalens 100, near-field information may be encoded into the propagatingwave by coupling to a passive waveguide.

Metalens 100 includes film 102 with subwavelength periodic cut-throughslits 108. In the exemplary embodiment, slits 108 are filled withdielectric elements 114, as described above in reference to FIG. 1.Dielectric elements 114 may be located within slits 108 using amulti-layer manufacturing process, in which alternating layers of film102 and dielectric elements 114 are formed on top of one another.Alternatively, dielectric elements 114 may be positioned within slits108 using any other suitable technique (e.g., implanting or injectingdielectric elements 114 into formed slits 108). For imaging purposes,dielectric elements 114 are composed of substantially opticallytransparent dielectric materials. Alternatively, slits 108 may be empty(i.e., unfilled). Slit width a and periodicity d are both in thesubwavelength regime (i.e., a<d<λ). Slits 108 extend through metalens100 in a direction substantially parallel to a direction in which theincident light is emitted from light source 1702. Periodic slits 108enable simultaneous multi-channel scanning, resulting in relatively fastimage acquisition.

As explained above, metalens 100 exhibits artificial dielectricbehaviors (e.g., enhanced non-linearity), and can be precisely mappedinto a homogeneous dielectric slab, showing a Fabry-Perot transmissionspectrum. At resonant frequencies, metalens 100 permits an extraordinarytransmission for a TE incoming wave (i.e., a wave with a transverseelectric field in the x direction). This transmission may potentiallyincrease the signal-to-noise ratio (SNR) for certain external noises.The extraordinary transmission results from the subwavelength periodicstructure, and cannot be achieved using only a single slit. Metalens 100may also be operated in an absorption mode.

Moreover, the electric fields in silts 108 are greatly enhanced (for aperfect electric conductor, the enhancement factor is d/a), and strongevanescent bulges 1710 are formed at both ends of slits 108, enhancinglight-matter interactions. The size of bulges 1710 is defined as thefull width at half maximum (FWHM) as compared with the electric field inslits 108. Numerically, the size of bulges 1710 is on the same order asslit width a. As explained above, the lateral resolution of an imageproduced using system 1700 is determined by slit width a, and may be oneor two orders of magnitude smaller than the wavelength λ of the incidentilluminating light. For example the resolution may be at least a factorof two smaller than the wavelength λ of the incident illuminating light.

To image object 1704, object 1704 is placed on the same side of metalens100 as light source 1702 such that object 1704 interacts with bulges1710 on that side of metalens 100. Detector 1706 is placed on theopposite side of metalens 100 and positioned in proximity of single slit108, referred to herein as a detection slit 1712. Detector 1706 measuresthe output optical flux, and accordingly the electric field strength,from detection slit 1712. Detector 1706 may be, for example, aphotodetector. In the embodiment shown in FIG. 17, only one detector1706 is shown. However, in some embodiments, system 1700 includes aplurality of detectors 1706, increasing a speed at which object 1704 maybe imaged. Further, in some embodiments, system 1700 includes one ormore optical fibers (not shown) that facilitating imaging by collectlight passing through metalens 100.

Notably, slits 108 are coupled such that when one slit 108 is covered byobject 1704, the field strengths in neighboring slits 108 are modifiedas well. Accordingly, each measurement by detector 1706 includes theinformation of the local optical properties of every periodic portion ofobject 1704 that covers a slit 108. Detector 1706 is communicativelycoupled to a computing device 1720. Computing device 1720 reconstructsimages of object 1704 based on the measurements acquired by detector1706, as described herein.

FIG. 18 is a schematic diagram for explaining data acquisition and imagereconstruction using system 1700. Assume object 1704 has an unknownrefractive index profile n(x) which is to be determined using system1700. The following describes determining n(x) over a continuous segment1802 of object 1704 with a length Nd, where N is an arbitrary integer.Further, the refractive index of object 1704 outside of segment 1802 isassumed to a uniform index n_(b).

Initially, a step size δ that specifies a pixel size in the image thatwill be reconstructed using system 1700 is selected. The imagingresolution is the greater of slit width a and step size δ. Segment 1802is mathematically divided into NΔ slices 1804, where Δ=d/δ. In theexemplary embodiment, metalens 100 is dimensioned such that δ=a and Δ isan integer. Accordingly, the width of each slice is a. Slices 1804 areindexed uniformly by n_(i) for i=1, 2, . . . NΔ. Slices 1804 are indexedfrom right to left as shown in FIG. 18.

Detector 1706 detects a first measurement E₁ with segment 1802positioned such that slice n₁ covers detection slit 1712. Notably,slices n_(1+Δ), n_(1+2Δ), . . . , n_(1+(N-1)Δ) also cover slits 108 whenslice n₁ covers detection slit 1712. The slices 1804 covering slits 108form a set S₁.

After first measurement E₁ is detected and recorded, object 1704 isshifted relative to metalens 100 by δ in the x direction. Accordingly,slice n₂ now covers detection slit 1712, and slices n_(2+Δ), n_(2+2Δ), .. . , n_(2+(N-1)Δ) cover other slits 108. These slices 1804 form a setS₂, and detector 1706 detects a second measurement E₂.

Shifting object 1704 and acquiring data is repeated until slice n_(NΔ)covers detection slit 1712 and measurement E_(NΔ) is recorded. In theexemplary embodiment, object 1704 is moved past metalens 100 at asubstantially constant velocity, and measurements are recorded whenslices 1804 align with slits 108. For example, object 1704 may becarried by a fluid that flows past object 1704 at a substantiallyconstant velocity. In some embodiments, measurements are taken at ahigher frequency (i.e., more often than only when slices 1804 align withslits 108), and those measurements taken when slices 1804 align withslits 108 are extracted from the remaining measurements for use in imagereconstruction. The extraction may be performed, for example, using datamasking. In an alternative embodiment, object 1704 does not move pastmetalens 100 at a constant speed, but is repeatedly shifted and stoppedat appropriate positions.

The NΔ measurements (i.e., E₁, E₂, . . . , E_(NΔ)) can be partitionedinto Δ groups according to their respective slice set (i.e., S₁, S₂, . .. , S_(Δ)). Each group contains N measurements. To extract informationabout segment 1802 from the measurements, an inverse differentialalgorithm reconstructs an image of segment 1802 from measurements asdescribed herein.

All of the measurements in each group are coupled. To extract n(x), itis assumed the segment 1802 being imaged is derived from a uniformmaterial with index n_(b), and a differential between measurements iscompared. A total field differential introduced by the sample is alinear summation of the contributions from all slits 108, and eachcontribution is proportional to a local index variation at every slit108. For example, the first measurement E_(j) in the j-th group can beexpressed as follows in Equation (2):E _(j) =E _(b)+(n _(j) −n _(b))C ₀ E _(inc)+(n _((j+Δ)) −n _(b))C ₁ E_(inc)+(n _((j+2Δ)) −n _(b))C ₂ E _(inc)+(n _((j+3Δ)) −n _(b))C ₂ E_(inc)+ . . .  (2)

In Equation (2), E_(inc) is the electric field of the incoming light,E_(b) is the field strength for a uniform sample with index n_(b), andC₁, C₂, C₃, . . . are dimensionless quantities that describe the selfand mutual couplings between detection slit 1712 and neighboring slits108, as shown in FIG. 2. The C_(i)'s can be either positive or negative.Within the j-th group, the N measurements can be recast into a compactmatrix form as follows in Equation (3):

$\begin{matrix}{{\begin{pmatrix}C_{0} & C_{1} & C_{2} & \ldots & C_{N - 1} \\C_{1} & C_{0} & C_{1} & \ldots & C_{N - 2} \\C_{2} & C_{1} & C_{0} & \ldots & C_{N - 3} \\\vdots & \vdots & \vdots & \ddots & \vdots \\C_{N - 1} & C_{N - 2} & C_{N - 3} & \ldots & C_{0}\end{pmatrix}_{N \times N}\begin{pmatrix}{\Delta\; n_{j}} \\{\Delta\; n_{({j + \Delta})}} \\{\Delta\; n_{({j + {2\Delta}})}} \\\vdots \\{\Delta\; n_{({j + {{({N - 1})}\Delta}})}}\end{pmatrix}_{N \times 1}} = \begin{pmatrix}{\Delta\;{\overset{\sim}{E}}_{j}} \\{\Delta\;{\overset{\sim}{E}}_{({j + \Delta})}} \\{\Delta\;{\overset{\sim}{E}}_{({j + {2\Delta}})}} \\\vdots \\{\Delta\;{\overset{\sim}{E}}_{({j + {{({N - 1})}\Delta}})}}\end{pmatrix}_{N \times 1}} & (3)\end{matrix}$

In Equation (3), Δ{tilde over (E)}_(j)=(E_(j)−E_(b))/E_(inc) andΔn_(j)=n_(j)−n_(b). Repeating the process for all groups provides n(x)for the entire segment 1802. Notably, rescaling all C_(i)'s does notchange a contrast of the reconstructed image. Moreover, in the exemplaryembodiment, metalens 100 is design such that only the first few C_(i)'scontribute. Through numerical simulation, it has been established thatselecting n_(b) based upon knowledge of object 1704 significantlyincreases the accuracy of the generated image. Moreover, the algorithmis also robust for a relatively wide range of n_(b).

FIG. 19 shows experimental results obtained using system 1700 and thereconstruction algorithm described herein. For the results shown in FIG.19, four samples 1902 of uniform silica (n=1.43869) each containing asingle Barium crown glass defect 1904 (n=1.54893) were imaged. Thethickness of each sample 1902 was 40 nm, and the wavelength of theincident light was 2.08 μm. Metalens 100 was made of silver(n=0.99886−14.128i) with slits 108 unfilled (i.e., empty), a periodicityof 400 nm, a slit width of 40 nm, and a length of 662 nm, such thatmetalens 100 was on resonance with the illuminating light. Numerically,for the example shown in FIG. 19, metalens 100 transmitted 66% of theincident light and reflects 3% of the incident light (the remaininglight dissipated). n_(b) was chosen to be the refractive index ofsilica.

The defect size, s, for the four samples 1902 was 60 nm, 40 nm, 20 nm,and 10 nm, respectively. A reconstructed image 1906 was generated foreach sample 1902. In the graphs shown in FIG. 19, the dashed linesrepresent the original index profile, and the dots connected by solidlines represent the reconstructed index profile. Reconstructed images1906 had FWHM of 70 nm, 62 nm, 58 nm, and 53 nm, respectively. For evensmaller defects, the FWHM numerically approach the size of light bulge1710, giving a point spread function (PSF) of system 1700.

As demonstrated in FIG. 19, system 1700 and metalens 100 can image afeature having a size smaller than slit width a, with the image blurredto have a size that is no less than slit width a. The kinks in thereconstructed profiles of the two larger samples 1902 and the peaks inreconstructed profiles of the two smaller samples 1902 are due to arelatively strong evanescent electric field in the y direction atcorners of slits 108. If object 1704 is moved slightly away (e.g.,approximately 10 nm) from metalens 100, these aberrations substantiallydisappear.

FIG. 20 shows experimental results obtained for attempting todistinguish a small distance, g, between two defects 2002 having size,s. Nine different samples 2004 with different combinations of distance gand size s were imaged to generate reconstructed images 2006. Samples2004 were silica with Barium crown glass defects 2002. The thickness ofeach sample 2004 was 40 nm, and the wavelength of the incident light was2.08 μm. Metalens 100 was made of silver with slits 108 unfilled, aperiodicity of 400 nm, a slit width of 40 nm, and a length of 662 nm,such that metalens 100 was on resonance with the illuminating light.

As demonstrated in FIG. 20, when g was less than or equal to 40 nm,defects 2004 were distinguishable from one another, regardless of theirsize. Further, a relatively small distance (e.g., g=20 nm) between tworelatively large defects (e.g., s=80, 60 nm) can be resolved by metalens100. Such a case can be considered as complementary to a case where arelatively small defect is embedded in an otherwise homogeneousmaterial. These results do not violate Abbe-Rayleigh criterion, whichdescribes the resolving power between two point objects.

Referring back to FIG. 17, placing detector 1706 in a near field ofmetalens 100 perturbs measurements acquired by detector 1706 to someextent. However, accurate image reconstruction is still possible insystem 1700, even in the presence of near field perturbation.

FIG. 21 is a schematic diagram of a waveguide 2102 positioned behindmetalens 100. Waveguide 2102 is positioned a distance, g_(w), from slit108 such that waveguide 2102 permeates a bulge 1710 associated with slit108. Waveguide 2102 also has a width, w_(w). Waveguide 2102 can perturbthe strong evanescent field, and excite propagating waves insidewaveguide 2102, which propagate through waveguide 2102 to be measured atan end of waveguide 2102 opposite bulge 1710.

FIG. 22 is a graph demonstrating that the reconstructed images with andwithout waveguide 2102 for a 1.2 μm long object 1704 are substantiallyin agreement with the original index profile. In the exemplaryembodiment, waveguide 2102 is made of silicon (n=3.44989) embedded in asilica substrate. Further, to facilitate maximizing a contrast to noiseratio, the width w_(w) of waveguide 2102 is 250 nm and the distanceg_(w) is 210 nm in the exemplary embodiment. Alternatively, waveguide2102 may have any composition and/or dimensions that enables waveguide2102 to function as described herein. In the exemplary embodiment, as aresult of the design, all C_(i)'s except C₀, C₁, and C₂ are equal tozero.

In imaging systems, noise may also contribute to a degradation in imagequality. Accordingly, the robustness of metalens 100 against externalrandom noise was evaluated. FIG. 23 shows experimental results for usingsystem 1700 to reconstruct images with and without noise in the presenceof waveguide 2102. Random noise was represented by 0.16%×I×C_(Rand),wherein I is the intensity of the incident light, and C_(Rand) is arandom number in the set [−1, 1].

In most biological tissues, due to light scattering and absorption,optical properties are described by a spatially varying complexrefractive index: ñ(x)=n(x)+ik(x). The correlations between slits 108allow ñ(x) to be determined in one scanning using two detectors 1706. Insuch a situation, when one slit 108 is covered by object 1704 with acomplex index, n(x) and k(x) contribute independently to the totalmeasured differential field (in general, n(x) and k(x) give rise to twodifferent sets of coupling constants C_(i)). By measuring at twoneighboring slits 108 (or in general, any two correlated slits 108),ñ(x) can be determined by applying the reconstruction algorithmdiscussed herein. Reconstructing the complex index ñ(x) based oncorrelations is not possible without multiple slits 108.

FIG. 24 is a block diagram of an exemplary computing device 2400, suchas computing device 1720 (shown in FIGS. 17 and 18), that may be usedwith system 1700.

Computing device 2400 includes at least one memory device 2410 and aprocessor 2415 that is coupled to memory device 2410 for executinginstructions. In some embodiments, executable instructions are stored inmemory device 2410. In the exemplary embodiment, computing device 2400performs one or more operations described herein by programmingprocessor 2415. For example, processor 2415 may be programmed byencoding an operation as one or more executable instructions and byproviding the executable instructions in memory device 2410.

Processor 2415 may include one or more processing units (e.g., in amulti-core configuration). Further, processor 2415 may be implementedusing one or more heterogeneous processor systems in which a mainprocessor is present with secondary processors on a single chip. Inanother illustrative example, processor 2415 may be a symmetricmulti-processor system containing multiple processors of the same type.Further, processor 2415 may be implemented using any suitableprogrammable circuit including one or more systems and microcontrollers,microprocessors, reduced instruction set circuits (RISC), applicationspecific integrated circuits (ASIC), programmable logic circuits, fieldprogrammable gate arrays (FPGA), and any other circuit capable ofexecuting the functions described herein. Processor 2415 processes datareceived from detector 1706 to reconstruct images of object 1704. In theexemplary embodiment, processor 2415 utilizes the reconstructionalgorithm described above to reconstruct images of object 1704.

In the exemplary embodiment, memory device 2410 is one or more devicesthat enable information such as executable instructions and/or otherdata to be stored and retrieved. Memory device 2410 may include one ormore computer readable media, such as, without limitation, dynamicrandom access memory (DRAM), static random access memory (SRAM), a solidstate disk, and/or a hard disk. Memory device 2410 may be configured tostore, without limitation, application source code, application objectcode, source code portions of interest, object code portions ofinterest, configuration data, execution events and/or any other type ofdata.

In the exemplary embodiment, computing device 2400 includes apresentation interface 2420 that is coupled to processor 2415.Presentation interface 2420 presents information to a user 2425. Forexample, presentation interface 2420 may include a display adapter (notshown) that may be coupled to a display device, such as a cathode raytube (CRT), a liquid crystal display (LCD), an organic LED (OLED)display, and/or an “electronic ink” display. In some embodiments,presentation interface 2420 includes one or more display devices.Reconstructed images of object 1704 may be displayed on presentationinterface 2420.

In the exemplary embodiment, computing device 2400 includes a user inputinterface 2435. User input interface 2435 is coupled to processor 2415and receives input from user 2425. User input interface 2435 mayinclude, for example, a keyboard, a pointing device, a mouse, a stylus,a touch sensitive panel (e.g., a touch pad or a touch screen), agyroscope, an accelerometer, a position detector, and/or an audio userinput interface. A single component, such as a touch screen, mayfunction as both a display device of presentation interface 2420 anduser input interface 2435.

Computing device 2400, in the exemplary embodiment, includes acommunication interface 2440 coupled to processor 2415. Communicationinterface 2440 communicates with one or more remote devices. Tocommunicate with remote devices, communication interface 2440 mayinclude, for example, a wired network adapter, a wireless networkadapter, and/or a mobile telecommunications adapter.

Embodiments provide a system for imaging an object using a metalens. Themetalens includes a two-dimensional array including a metallic filmhaving periodic slits. The metalens exhibits artificial dielectricbehaviors and can be precisely mapped into a homogeneous dielectricslab. Further, the periodic structure results in a relatively hightransmission of incident light. Using the metalens, reconstructed imageshave a resolution much smaller than a wavelength of the incident light.

The order of execution or performance of the operations in theembodiments of the disclosure illustrated and described herein is notessential, unless otherwise specified. That is, the operations may beperformed in any order, unless otherwise specified, and embodiments ofthe disclosure may include additional or fewer operations than thosedisclosed herein. For example, it is contemplated that executing orperforming a particular operation before, contemporaneously with, orafter another operation is within the scope of aspects of thedisclosure.

When introducing elements of aspects of the disclosure or embodimentsthereof, the articles “a,” “an,” “the,” and “said” are intended to meanthat there are one or more of the elements. The terms “comprising,”including,” and “having” are intended to be inclusive and mean thatthere may be additional elements other than the listed elements.

This written description uses examples to disclose the disclosure,including the best mode, and also to enable any person skilled in theart to practice the disclosure, including making and using any devicesor systems and performing any incorporated methods. The patentable scopeof the disclosure is defined by the claims, and may include otherexamples that occur to those skilled in the art. Such other examples areintended to be within the scope of the claims if they have structuralelements that do not differ from the literal language of the claims, orif they include equivalent structural elements with insubstantialdifferences from the literal language of the claims.

What is claimed is:
 1. A method for imaging an object, said methodcomprising: positioning the object between a light source and ametalens, wherein the metalens includes a metallic film having aplurality of slits defined therethrough, the plurality of slits having awidth a and a periodicity d; emitting light from the light sourcetowards the object, wherein the emitted light has a predeterminedwavelength greater than the width a and the periodicity d; acquiring,using a detector, measurements indicative of light transmitted throughthe metalens; and reconstructing, using a computing devicecommunicatively coupled to the detector, an image of the object based onthe acquired measurements.
 2. A method in accordance with claim 1,wherein positioning the object comprises positioning the object suchthat at least a portion of the object lies within evanescent bulgesformed at ends of the plurality of slits.
 3. A method in accordance withclaim 1, further comprising locating a plurality of dielectric elementswithin the plurality of slits.
 4. A method in accordance with claim 3,wherein locating a plurality of dielectric elements comprises locating aplurality of substantially optically transparent dielectric elements. 5.A method in accordance with claim 1, wherein reconstructing an imagecomprises reconstructing an image having a resolution at least a factorof two smaller than the predetermined wavelength.
 6. A method inaccordance with claim 1, wherein the detector is positioned proximate adetection slit of the plurality of slits.
 7. A method in accordance withclaim 1, further comprising positioning a waveguide between the metalensand the detector.